Form Of Convolution Integral Research Articles

Analytical solution of the boundary value problem of mathematical modeling for non-stationary oil flows through the trunk pipeline in the presence of internal pressure sources

The main oil pipeline, due to its spatial extent, can be considered as a control object with distributed parameters (ODP). The dependences on time and coordinates of the flow velocity and pressure in the pipeline are considered as controlled output values of the ODP. The boundary value problem of mathematical modeling of the process of pipeline transportation of oil in the standard form is presented in the form of a linear partial differential equation of the second order. The paper presents a solution to the boundary value problem of mathematical modeling of unsteady oil flow through a trunk pipeline in the presence of internal concentrated pressure sources in the form of functions describing the dependences on time and spatial coordinates of pressures and average cross-section pipeline oil flow rates. To represent the solution of the boundary value problem in the form of convolution integrals, Green's functions and standardizing functions are obtained, which makes it possible to use non-smooth (discontinuous) dependencies to describe programs for changing the values of internal concentrated pressure sources over time. The solutions obtained make it possible to use the methods of the theory of optimal control of systems with distributed parameters to solve the problems of optimal control of the process of pipeline transportation of oil.

The Transient Motion of A Current Loop Along A Conducting Cylindrical Shell.(Dept.E)

A mathematical and an analytical procedure for calculating the induced surface eddy currents in a cylindrical shell due to an arbitrary moving concentric current loop is developed. The consideration involves both uniform and accelerated motions. The accelerated motion, has the form of a finite ramp function, by which the current loop moves with a constant acceleration during a given time interval to a certain velocity. The surface currents induced in the cylindrical shell are governed by a time-dependent linear differential equation, which is then solved by the method of Laplace transforms. A solution in the form of Fourier and convolution integrals is achieved. The analytical solution is demonstrated in a graphical form in which the induced surface currents for both uniform and accelerated motion are presented. The solution so obtained is then extended to an arbitrary motion. The force acting on the moving current loop is also calculated and the results are presented in analytical and graphical forms.

Fresnel diffraction of multiple disks on axis

Aims. We seek to study the Fresnel diffraction of external occulters that differ from a single mask in a plane. Such occulters have been used in previous space missions and are planned for the future ESA Proba 3 ASPIICS coronagraph. Methods. We studied the shading efficiency of double on-axis disks and generalized results to a 3D occulter. We used standard Fourier optics in an analytical approach. We show that the Fresnel diffraction of two and three disks on axis can be expressed using a Babinet-like approach. Results are obtained in the form of convolution integrals that can be written as Bessel-Hankel integrals; these are difficult to compute numerically for large Fresnel numbers found in solar coronagraphy. Results. We show that the shading efficiency of two disks is well characterized by the intensity of the residual Arago spot, a quantity that is easier to compute and therefore allows an interesting parametric study. Very simple conditions are derived for optimal sizes and positions of two disks to produce the darkest structure around the Arago spot. These conditions are inspired from empirical experiments performed in the sixties. A differential equation is established to give the optimal envelope for a multiple-disk occulter. The solution takes the form of a simple law, the approximation of which is a conical occulter, a shape already used in the SOHO Mission. Conclusions. In addition to quantifying expected results, the present study highlights unfortunate configurations of disks and spurious diffractions that may increase the stray light. Particular attention is paid to the possible issues of the future occulter spacecraft of ASPIICS.

Dynamic Force Identification Problem Based on a Novel Improved Tikhonov Regularization Method

The main purpose of this paper is to identify the dynamic forces between the conical pick and the coal-seam. According to the theory of time domain method, the dynamic force identification problem of the system is established. The direct problem is described by Green kernel function method. The dynamic force is expressed by a series of functions superposed by impulses, and the dynamic response of the structure is expressed as a convolution integral form between the input dynamic force and the response of Green kernel function. Because of the ill-conditioned characteristics of the structure matrix and the influence of measurement noise in the process of dynamic force identification, it is difficult to deal with this problem by the usual numerical method. In present content, a novel improved Tikhonov regularization method is proposed to solve ill-posed problems. An engineering example shows that the proposed method is effective and can obtain stable approximate solutions to meet the engineering requirements.

Resonance responses in a two-degree-of-freedom viscoelastic oscillator under randomly disordered periodic excitations

An investigation is presented for primary resonance and internal resonance of two-degree-of-freedom (TDOF) viscoelastic system with some complex nonlinear coupled terms under randomly disordered periodic excitations. The assumed viscoelastic damping depending on the past history of motion is chosen as the form of convolution integrals over an exponentially decaying kernel function. Then, the steady-state responses of the TDOF system are discussed in detail through the method of multiple scales (MMS), which is employed to derive the determined and stochastic differential equations of amplitude and phase modulation for the internal and external resonance modes. Further, the numerical simulation method is absorbed to test the effectiveness and accuracy of the theoretical analysis solutions for steady-state moments with the changing excitation amplitude. The appearance of (random) jump, (random) saturation, and double-jumping can be explored by the development of steady-state moments with different excitation amplitudes and frequency, which also show the resonance bandwidth increases with the excitation amplitude. Besides, viscoelastic damping parameters have great influence on the steady-state responses under the different detuning conditions, which accelerate or delay the appearance of the above mentioned phenomena. Numerical simulation results of phase portraits tell that the increase of the intensity of the random excitation leads to the steady-state solutions changing from a limit cycle to a diffused limit cycle, thus, such multi-valued steady-state responses produce jump from one stable branch to another also response to the previous analysis conclusion.

A Thermo-Poro-Viscoelastic Model for the Behavior of Polymeric Porous Separators

Porous separator membranes play an important role in the long-term performance of lithium battery cells. They provide a barrier between two electrodes and enable migration of lithium ions through the pores, where lithium ions migrate through the electrolyte via diffusion-migration and reach the electrode materials. During the discharge process, the intercalation of Li ions into active materials in the electrode leads to swelling of active particles, expansion of the electrode, mechanical stresses, and compression of the separator membrane. Moreover, the extreme thermal condition lessens the performance of the separator by closing the pores through a melting process. To assess the performance of the separator under mechanical and thermal conditions, we present a thermo-poro-viscoelastic model for polymeric porous separators. We study the thermo-mechanical behavior of fully saturated porous separators through the thermo-poromechanics formulations. Two-phase mixture with solid and fluid phases is considered. Governing and balance equations are presented for the solid-fluid mixture, where the displacement of the solid skeleton, temperature and fluid pore pressure are considered as unknowns with a quasi-static loading condition. The stress-strain behavior of the polymeric separator membrane is described by a viscoelastic constitutive law in the form of convolution integral, where the thermal effects are considered through the thermal strain. The flow of the fluid through the porous medium is described by the Darcy’s law. The constitutive equation for the Darcy fluid velocity is described by the permeability of the fluid. To present more realistic physical phenomena, the hydraulic conductivity is described as a function of the porosity in the solid phase. Numerical simulations are performed to study the performance of the separator in a saturated porous medium.

Elegant deduction of three types of recursive convolution (RC) technique in simulation of dispersive material

Recursive convolution (RC) method is widely used in simulation of dispersive material in finite-difference time-domain (FDTD). The method uses inverse Fourier transform to change the constitutive relation between polarization and electric field in frequency-domain to convolution integral form in time domain. The convolution is updated by the recursive method. This paper presents the elegant deduction of the RC method and shows that constant RC (CRC), piecewise linear RC (PLRC) and trapezoidal RC (TRC) are all special cases of a general form. The difference is due to the approximation of electric field in convolution integral. The numerical examinations validate the accuracy and efficiency of the algorithms.

Modeling of Quasistatic Thermoviscoelastic Frictional Contact Problems

Frictional contacts of thermoviscoelastic bodies are complicated nonlinear temperature- and time-dependent problems. The introduction of friction with its irreversible character makes the problem more difficult. Additionally, the consideration of temperature, as an independent variable, destroys the convolution integral form of the viscoelasticity constitutive relations. This paper presents a computational model capable of predicting the nonlinear quasistatic response of uncoupled thermoviscoelastic frictional contact problems. The contact problem, as a variational inequality constrained model, is handled by using the Lagrange multiplier method to incorporate the inequality contact constraints. A local nonlinear friction law is adapted to model friction at the contact interface. This, in turn, eliminates difficulties that arise with the application of the classical friction laws. The temperature-dependency of viscoelasticity is modeled by applying the time-temperature superposition principle. The constitutive equations are transformed to be a function of the reduced time as the only independent variable, maintaining the convolution integral form. Two different illustrative examples are presented to demonstrate the applicability of the proposed model to analyze both nonconformal and conformal thermoviscoelastic frictional contact problems.

Nonlocal finite element modeling of the tribological behavior of nano-structured materials

A nonlocal finite element model is developed for solving elasto-static frictional contact problems of nanostructures and nanoscale devices. A two dimensional Eringen-type nonlocal elasticity model is adopted. The material is characterized by a stress-strain constitutive relation of a convolution integral form whose kernel is capable to take into account both the diffusion process of nonlocal elasticity and the scale ratio effects. The incremental convex programming procedure is exploited as a solver. Two examples of different nature are presented, the first one presents the behavior of a nanoscale contacting system and the second example discusses the nano-indentation problem.

Numerical computation in the viscoelastoplastic continuum damage of hot mix asphalt concrete

ABSTRACTThe objective of this paper is to develop an accurate and advanced material characterization of hot mix asphalt concrete using an existing viscoelastoplastic constitutive model that accounts for rate of loading, temperature and stress state with growing damage. The modelling strategy of viscoelastoplastic continuum damage is based on modelling strain components separately and then combining the resulting models to obtain a final integrated viscoelastoplastic model. According to this model, the initial‐boundary value problem is numerically solved using the constitutive relationship expressed in the convolution integral form. The model is successful in predicting responses up to localization when microcracks start to coalesce.

A quasistatic analysis for thermoviscoelastic contact problems

Thermomechanical contact of viscoelastic bodies is a non-linear time- and temperature-dependent problem. Consideration of temperature as an independent variable destroys the convolution integral form of the viscoelasticity constitutive relations. This paper presents a computational procedure capable of predicting the quasistatic response of uncoupled, thermoviscoelastic, frictionless contact problems. The contact problem as a convex programming model is solved throughout an incremental procedure. The Wiechert model is adopted to simulate the linear behaviour of viscoelastic materials. The temperature dependency of viscoelasticity is accounted for by applying the time—temperature superposition principle, in which the William, Landel, and Ferry relationship is adopted to determine the shift factor. Thus, the constitutive equations are transformed to be a function of the reduced time as the only independent variable, maintaining the convolution integral form. Therefore, the complications that arise during the direct integration of these equations, especially with contact problems, are avoided. Two different illustrative examples are included to demonstrate the applicability of the proposed procedure.

TEMPORAL CAVITY OSCILLATIONS CAUSED BY A WIDE-BAND WAVEFORM

Excitation of the electromagnetic fields by a wide-band current surge, which has a beginning in time, is studied in a cavity bounded by a closed perfectly conducting surface. The cavity is filled with Debye or Lorentz dispersive medium. The fields are presented as the modal expansion in terms of the solenoidal and irrotational cavity modes with the time-dependent modal amplitudes, which should be found. Completeness of this form of solution has been proved earlier. The systems of ordinary differential equations with time derivative for the modal amplitudes are derived and solved explicitly under the initial conditions and in compliance with the causality principle. The solutions are obtained in the form of simple convolution (with respect to time variable) integrals. Numerical examples are exhibited as well.

Numerical Simulation and Characterization of Press Molding of Thermo-Plastic Resin Reinforced by Carbon Nanofiber

In the present paper, press molding for thermo-plastic resin toughened by carbon nanofiber were discussed. Polycarbonate (PC) and vapor grown carbon nanofiber (VGCF) are used for the resin matrix and the reinforcement. In order to investigate and simulate the process molding process of the PC/VGCF composite by finite element method, the thermo-viscoelastic property of the composites are estimated using unidirectional compression creep test. The creep functions were converted into relaxation function based on the convolution integral form of the basic equation using Laplace transform. Experimental testing for press molding of the PC/VGCF composites using V-shape die has been carried out. The results of the V-shape profiles and those obtained by FEM analysis were compared to confirm the validity of the present analysis method based on thermo-viscoelastic theory. The effect of the carbon nanofiber for the formability improvement of the PC/VGCF composites was discussed in detail.

Constitutive modeling of porous viscoelastic materials

The effect of porosity on the constitutive response of an isotropic linearly viscoelastic solid that obeys a constitutive law of the standard differential form is investigated under small strain deformation conditions. The correspondence principle of linear viscoelasticity is used to solve the viscoelastic boundary value problem at a unit cell containing a spherical void and loaded axisymmetrically by macroscopic stresses. The results are used to devise a constitutive potential for the description of the porous material for any arbitrary combination of hydrostatic and deviatoric loadings, and the associated 3-D constitutive relationship is determined in the Laplace transform domain. Inversion to the time domain yields the constitutive law of the porous material as a function of porosity in the standard form of convolution integrals. The presence of porosity establishes relaxation time scales for the porous body that differ from the relaxation time of the pure matrix material and brings about a viscous character to the overall hydrostatic response. The numerical implementation of the model in a general purpose finite element code is outlined. The model is used to predict the response of a porous solid propellant material in uniaxial tension and cyclic loading at room temperature.

Numerical computation of convolution integral for the linear viscoelasticity of asphalt concrete

This study is concerned with the development of a numerical formulation for the convolution integral which represents a current time response from all previous time history in asphalt concrete. To solve the initial/boundary value problem involving linear viscoelasticity, the constitutive equation, expressed in the convolution integral form connected with the relaxation modulus obtained from frequency sweep tests of asphalt concrete specimens, is numerically formulated into an incremental expression. The relaxation modulus can be resulted in the mechanical models consisting of linear springs and dashpots. The mechanical Wiechert model of relaxation modulus is formulated by Prony series used for the numerical integral formulation based on the linear viscoelasticit y. For evaluation purpose, numerical examples involving a crosshead tension (or a constant-strain-rate-monotonic loading), a saw tooth shape loading, and a beam under tip load are incorporated into the validation of the numerical integration.

上下限領域に閉じ込めたParticle Swarm Optimizationの力学系の分岐特性と収束特性

In this paper, we propose two types of dynamics of Particle Swarm Optimization (PSO) embedded into upper and lower bounds, which have more powerful abilities of searching the global optimal solution on their bounded constraints. The new type of dynamics called nonlinear operator and nonlinear variable transformation are inertial models, which are introduced by discretizing the convolution integral form on continuous time. After certifying stability and/or instability including chaos of dynamics by the bifurcation diagrams, the infuences of model parameters settings on the global search are investigated to achieve better performance than the orthodox PSO dynamics in copmutational simulations for a few benchmark objective functions.

Magnetic reconnection with space and time varying reconnection rates in a compressible plasma

Fast magnetic reconnection of Petschek-type including moving shock waves and discontinuities in a compressible plasma is studied. Magnetic flux tubes of finite size are reconnected by a localized dissipative electric field pulse. This process generates nonlinear perturbations propagating along the initial current surface. The linear wave problem in the outer regions is solved analytically in terms of the reconnection induced sources which move in different directions and with different speeds along the surface. The time-coordinate representation of the solution is given in form of convolution integrals over the reconnection initializing electric field. As an example, reconnection of flux tubes in a sheared magnetic field geometry is analyzed.

Inversion of loading time history using displacement response of composite laminates: Three-dimensional cases

An inverse procedure is proposed to reconstruct the time history of transient loads on the surface of composite laminates from the knowledge of dynamic displacement response at only one receiving point. A hybrid numerical method (HNM) is adopted as the forward solver to compute the dynamic displacement response of composite laminates subjected to arbitrary loads. By introducing a kernel displacement function — the dynamic displacement response of composite laminates excited by a point stepimpact load and the displacement response subjected to a load with an arbitrary force function are expressed in a form of convolution integral. The force history is reconstructed by employing an inversion algorithm, in which the least-squares optimization method being adopted to deconvolute the integral. Both point loads and loads with small spatial distribution are investigated and numerical verifications are given. The robustness of the procedure in the presence of noise is also investigated. Good agreements between the identified and true functions for all cases demonstrate the effectiveness of the present inverse procedure. The present inverse procedure is useful for determining impact loads on material surface using response on a point remote to the impact point.

A Fast and Effective Method for Transient Thermoelastic Displacement Analyses

Frictional heating due to the relative motion of contacting surfaces causes temperature rise and thermal distortion, which in turn affects the contact geometry and pressure distribution. A fast and effective method is presented for the calculation of the normal surface displacement of an elastic halfspace due to arbitrary transient surface heating. The method uses Fourier-transformed Green’s functions (frequency response functions), found in the closed form by using the approach of Seo and Mura and the heat conduction analyses of Carslaw and Jaeger. The frequency response functions are shown analytically to be the frequency domain representations of the Green’s functions given by Barber. The formulation for the surface normal displacement is in the form of three-dimensional convolution integrals (over surface and time) of the arbitrary transient heat flux and the Green’s functions. Fourier transforms of these convolution integrals are taken, avoiding the Green’s-function singularities and giving a simple multiplication between the transformed heat flux and the (known) frequency response functions. The discrete convolution–fast Fourier transform (DC-FFT) algorithm is applied for accurate and efficient calculations of the normal surface displacement from the frequency response functions for an arbitrary transient heat input. The combination of the frequency-domain formulation and the DC-FFT algorithm makes the solution of transient thermoelastic deformation extremely fast and convenient.

Damping described by fading memory—analysis and application to fractional derivative models

Some damping models where the actual stress does not depend on the actual strain but also on the entire strain history are studied. Basic requirements in the frequency and time domain significant for the choice of damping model are outlined. A one-dimensional linear constitutive viscoelastic equation is considered. Three different equivalent constitutive equations describing the viscoelastic model are presented. The constitutive relation on the convolution integral form is studied in particular. A closed form expression for the memory kernel corresponding to the fractional derivative model of viscoelasticity is given. The memory kernel is examined with respect to its regularity and asymptotic behavior. The memory kernels relation to the fractional derivative operator is discussed in particular and the fractional derivative of the convolution term is derived. The fractional derivative model is also given by two coupled equations using an internal variable. The inclusion of the fractional derivative constitutive equation in the equations of motion for a viscoelastic structure is discussed. We suggest a formulation of the structural equations that involves the convolution integral description of the fractional derivative model of viscoelasticity. This form is shown to possess several mathematical advantages compared to an often used formulation that involves a fractional derivative operator form of constitutive relation. An efficient time discretization algorithm, based on Newmarks method, for solving the structural equations is presented and some numerical examples are given. A simplification of the fractional derivative of the memory kernel, derived in the present study, is then employed, which avoids the actual evaluation of the memory kernel.